Hi, folks,I'm learning calculus off an online course (http://www.math.byu.edu/~smithw/Calculus/), and I have a question. On page 8 of chapter 4 (http://www.math.byu.edu/~smithw/Calculus/Differential-Chapter4/) theorem D7, it goes into how to calculate e. At one point, it gets to e**(x/e**(x)-1)). Up until that point I'm ok. The next step is, ((e**x)-1)/x =1 as x->0 since (e**x)' as x->0 is (e**0)' = e**0 = 1. But how did they get from one step to the next? e is first being raised to x/((e**(x-1)), and now they write, (e**(x-1))/x? They seem to be using it to prove that (e**(x-1))/x goes to 1 as x->0, but how does that proof work? How do they get from one step to the next?Anyway, thanks for any help you can give. It may be clearer just to click on the links. I'd cut and paste, but the thing was scanned onto the internet, as opposed to typed.

Thanks again, SuperShuki

**=raised to the power of

_________________“Once you have tasted flight, you will forever walk the earth with your eyes turned skyward, for there you have been, and there you will always long to return.” -Anonymous

Thanks JamesThere are page numbers, and on the eighth page (8. in the upper left of the page) It says, "exponential functions, hyperbolic functions". That's where it starts talking about e. At the bottom of the screen, you should see Theorem D7. The second paragraph after that says Theorem: calculation of e. Go past all the exercises and the review questions and the I series and it should be right there.

Thanks again for your time,SuperShuki

_________________“Once you have tasted flight, you will forever walk the earth with your eyes turned skyward, for there you have been, and there you will always long to return.” -Anonymous

Last edited by SuperShuki on Wed Jun 12, 2013 11:11 pm, edited 1 time in total.

(Ignore the dots in the first column, the forum software removes leading whitespace, messing up my formatting .)

Replacing x with 0 in the part of the above equation on the right, and then h with x (!), we get

Code:

. (0+x) 0 x. e - e e - 1. ------------- = -------. x x

So, as x (used to be h) goes to zero, this converges to the derivative at 0 (since we replaced x with zero), which equals 1. The inverted fraction x / (e^x - 1) must then also go to 1 as x goes to zero, and the rest follows.

The variable substitution is a bit confusing, and it would have been nice if he'd given a warning or a bit more explanation. Anyway, I hope that that clarifies it.

_________________Say, can you feel the thunder in the air? Just like the moment ’fore it hits – then it’s everywhereWhat is this spell we’re under, do you care? The might to rise above it is now within your sphereMachinae Supremacy – Sid Icarus

Well, you can rest assured that Leonhard Euler died before Karl Marx was born, so the number e is completely free of communist influence and can safely be used by those of the anti-social persuasion .

_________________Say, can you feel the thunder in the air? Just like the moment ’fore it hits – then it’s everywhereWhat is this spell we’re under, do you care? The might to rise above it is now within your sphereMachinae Supremacy – Sid Icarus

I'll leave the check for x = ln(3/2 + sqrt(5)) as an exercise for the reader .

_________________Say, can you feel the thunder in the air? Just like the moment ’fore it hits – then it’s everywhereWhat is this spell we’re under, do you care? The might to rise above it is now within your sphereMachinae Supremacy – Sid Icarus

Last edited by Lourens on Sun Sep 22, 2013 10:14 pm, edited 1 time in total.

There has to be a typo somewhere, because what you posted reduces to x = y, not x^2 - y^2 = 4.

_________________Say, can you feel the thunder in the air? Just like the moment ’fore it hits – then it’s everywhereWhat is this spell we’re under, do you care? The might to rise above it is now within your sphereMachinae Supremacy – Sid Icarus

_________________Say, can you feel the thunder in the air? Just like the moment ’fore it hits – then it’s everywhereWhat is this spell we’re under, do you care? The might to rise above it is now within your sphereMachinae Supremacy – Sid Icarus